Cory, yes, it appears you're reading that correctly. Most of the WordPress bundled themes use this style because it's a cleaner way to do it. In this case, you want to focus on the get_template_part section near the bottom of the code you pasted.

See where it says get_template_part( 'content', 'single' ); ? That's referencing a second file called content-single.php, or content.php or single.php (confusing, right?). I suggest you look in your file list in the editor and try to locate a file called content-single.php (or, if it doesn't exist, one of the other choices I listed).

If you're interested, you can read more about get_template_part in the Codex article.

Is there a plugin that simply removed H1 and H2 defualt tags on single posts.

Not that I've ever seen.

Unfortunately, it looks like the markup you seek might live inside the theme_post_wrapper() function which can probably be found in your theme's functions.php file. I frankly can't fathom why the theme developer would do it this way ... it makes it incredibly difficult to customize after the fact as I'm sure you're well aware by now :)

From what I'm seeing, it looks like the developer has wrapped everything up in functions. You might take another close look at the single.php. Look above the Loop section, e.g. if ( have_posts() ... and see if there's anything to do with the_title() or get_the_title().

FYI, when you're going to be posting big blocks of code, you should use http://wordpress.pastebin.com instead of posting it directly here. You just paste the code there, click submit then supply the URL to it here. Sometimes wayward code can wreak havoc on the forums :)

If you have FTP access, I suggest logging in and replacing the file with the original (if you have it or can get it). My best suggestion at this point is to approach whoever developed your theme and asking them what the best way forward is for customizing your single page. It's getting late here, I'm just sorry I could help you get your (original) problem resolved.